Deleted emails in Mail keep coming back since 10.10.3 update

Question

Mac Mail won't let me delete email or move email to other mailboxes since updating to 10.10.3. I delete messages or move them to other folders/mailboxes, and they keep coming back to my Inbox. Some messages come back almost instantly. Some come back after a few minutes. Other messages have come back that I deleted earlier in the day. There don't appear to be duplicates, since the messages I deleted aren't in the trash or in the folder/mailbox to which I moved them. Some messages have stayed in the trash, but when I try to permanently delete them, they reappear. The only way I've been able to permanently delete messages is by going online and deleting them in Gmail. I have used Gmail in combination with Mail for years without problems, and I had been using an earlier version of Yosemite without problems. My mail account is a Gmail account set up as IMAP. I have folders/mailboxes that are in Gmail and I have folders/mailboxes that are on my computer, but the messages I move won't stay in either type of mailbox. I tried rebuilding the mail envelop with Onyx which has helped with Mail quirks in the past, but this time it did nothing to solve the problem.
Macbook Pro, Mid 2010, 2.66 GHz, Intel Core i7

Answer 1

I have an android phone. If I read a message on my phone, it still shows up on my desktop.
One of the problems with deleting messages is that I when I delete messages from the main Inbox (the Inbox that includes all incoming messages from all accounts), some of those messages stay in the Inbox for the individual account as unread. A typical scenario is that after reading all of my new email and deleting messages, I notice that Mail shows that I still have unread mail. So I go to the Inbox for an individual email account, and there are a bunch messages that I deleted (sometimes half, sometimes all of what I deleted) that are in the Inbox of the individual account and show as unread. Then when I go back to the main Inbox, and all of the messages that I deleted are suddenly back in the that Inbox. There doesn't seem to be any pattern to which messages get stuck. Sometimes the messages go away on their own. Sometimes they won't ever go away. I delete them from the main Inbox; they come back. I delete them from the individual account Inbox; they come back. I go to the Trash, delete them before they've gone back the the Inboxes, and they come back. The only way I can get rid of these zombie messages is to access the account through a browser and delete those messages online.
Another problem I have, is that Mail is constantly asking me to re-enter the password for one of my email accounts. Fortunately, it's an account I use less frequently, but it's a real problem when I Mail asks me to re-setup up an account every time I open Mail.
The problem has doubled or tripled the time it takes for me to keep track of my email. I'm going through reviews and looking for a replacement for Mail. I will be the first time I've used something besides Mail since I used Eudora (during the time when it was the best program available and you had to buy it).